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Multi-fusion categories are a generalization of fusion categories with a non-simple unit. The direct sum of two multi-fusion categories is again a multi-fusion category. By irreducible I mean that a multi-fusion category cannot be written as a non-trivial direct sum. Are all such irreducible multi-fusion categories fusion categories?

Bonus question: Are there multi-fusion categories that are not Morita equivalent to a direct sum of fusion categories?

[I'm mainly interested in the unitary case.]

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Matrix categories, $\mathrm{End}(\mathrm{Vec}^{\oplus n})$. (The identity on each copy of $\mathrm{Vec}$ are summands of the identity.)

(You can generalize this example by putting fusion categories along the diagonal of a matrix, and Morita equivalences between them into the off-diagonal entries. Over an algebraically closed field irreducible multifusion categories examples look like this, which you can see by decomposing the multifusion category as a module over itself.)

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    $\begingroup$ Is the implicit (sub)text of this answer "Matrix categories are irreducible multi-fusion categories that are not fusion categories"? $\endgroup$
    – LSpice
    Commented Mar 4, 2019 at 18:36
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    $\begingroup$ Indeed, matrix categories are irreducible multi-fusion categories that are not fusion categories. They cannot be written as direct sums of other multi-fusion categories. $\endgroup$
    – André Henriques
    Commented Mar 4, 2019 at 19:58
  • $\begingroup$ Oh, right of course. I wanted to answer the physics question "are there non-symmetry-breaking topological phases arising from multi-fusion categories that go beyond the Levin-Wen/Turaev-Viro model for fusion categories?", but your answer shows that what I asked is not equivalent to this, as your matrix-category example is still a trivial phase. $\endgroup$
    – Andi Bauer
    Commented Mar 6, 2019 at 18:27
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    $\begingroup$ Assuming the base field is C (weird issues arise if it's not algebraically closed), I'm pretty sure any multifusion category is Morita equivalent to a fusion category. Take $1_i$ to be a summand of $1$, and look at the category of $1_i$-modules. This gives a Morita equivalence between your original category and the fusion category of $1_i$-bimodules. This is a rephrasing of my parenthetical. $\endgroup$
    – Noah Snyder
    Commented Mar 6, 2019 at 19:04
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    $\begingroup$ Ok thanks! But shouldn't it rather be "any irreducible multifusion category is Morita equivalent to a fusion category"? Because a direct sum of fusion categories can never be Morita equivalent to a fusion category. Or am I confusing something here? $\endgroup$
    – Andi Bauer
    Commented Mar 7, 2019 at 14:39

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